This vector has two critical points of index \(+1\) , each when the radial unit vector points along the axis of the dipole: \({e}_{r}({e}_{r}\cdot {p})={p}\) . We can thus see that the nodes in the radiation pattern from a simple dipole—something we are familiar with from a first course in electromagnetism and shown in Fig. REF —can be understood to be a necessary consequence of the topology of a sphere. Were the dispersion surface to have a different topology (in a hyperbolic material, for example [1]}), these properties would change. This line of argument has far reaching consequences for the polarization of scattered light, governing the polarization of sunlight [2]} and multipole scattering in metamaterials [3]}.
